Counting the number of isotone selfmappings of crowns

Letn>1. The number of all strictly increasing selfmappings of a 2n-element crown is . The number of all order-preserving selfmappings of a 2n-element crown is

Extensions of hereditary algebras

Let be a triangular matrix algebra, uhere k is an algebraically closed field, B is the path algebra of an oriented Dynkin diagram of type E₆ or E₇ or E₈ and M is a finite dimensional k-B-bimodule. The aim of this paper is to determine the representation type of A for any orientation of the Dynkin diagram and for any indecomposable B-module M. This classification is obtained by comparing the representation types of the algebras and using the theory of tilting modules.     

Solutions of minimal period for a semilinear wave equation

In questo lavoro si considera il problema

Branching inset entropies on open domains

The forms of all semisymmetric, branching, multidimensional measures of inset information on open domains are determined. This is done for both the complete and the possibly incomplete partition cases. The key to these results is to find the general solution of the functional equation

Eigenvalues of the Dirac operator in the continuous spectrum

Functionsp(x) andq(x) for which the Dirac operator $$Dy = \left( {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ { - 1} \\ \end{array} } & {\begin{array}{*{20}c} 1 \\ 0 \\ \end{array} } \\ \end{array} } \right)\frac{{dy}}{{dx}} + \left( {\begin{array}{*{20}c} {p(x) q(x)} \\ {q(x) - p(x)} \\ \end{array} } \right)y = \lambda y, y = \left( {\begin{array}{*{20}c} {y_1 } \\ {y_2 } \\ \end{array} } \right), y_1 (0) = 0,$$ has a countable number of eigenvalues in the continuous spectrum are constructed.     

On theq-log-concavity of Gaussian binomial coefficients

We give a combinatorial proof that is a polynomial inq with nonnegative coefficients for nonnegative integersa, b, k, l withab andlk. In particular, fora=b=n andl=k, this implies theq-log-concavity of the Gaussian binomial coefficients , which was conjectured byButler (Proc. Amer. Math. Soc. 101 (1987), 771–775).     

Gauss polynomials and the rotation algebra

Newton's binomial theorem is extended to an interesting noncommutative setting as follows: If, in a ring,ba=ab with commuting witha andb, then the (generalized) binomial coefficient arising in the expansion

Estimating the diameter of one class of functions in L2

Let

The formula for the number of order-preserving selfmappings of a fence

LetY be a fence of sizem andr=?m?1/2?. The numberb(m) of order-preserving selfmappings ofY is equal toA _r-B_r-C_r-D_r, where, ifm is odd, $$\begin{gathered} A_r = 2(r + 1)\sum\limits_{s = 0}^r {\left( {\begin{array}{*{20}c} {r + s} \\ {2s} \\ \end{array} } \right)} 4^s , B_r = 2r\sum\limits_{s = 1}^r {\left( {\begin{array}{*{20}c} {r + s} \\ s \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {r - 1} \\ {s - 1} \\ \end{array} } \right),} \hfill \\ C_r = 4r\sum\limits_{s = 0}^{r - 1} {\left( {\begin{array}{*{20}c} {r + s} \\ s \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {r - 1} \\ s \\ \end{array} } \right), D_r = \sum\limits_{s = 0}^{r - 1} {(2s + 1)} \left( {\begin{array}{*{20}c} {r + s - 1} \\ s \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {r - 1} \\ s \\ \end{array} } \right)} \hfill \\ \end{gathered} $$ . Ifm is even, a similar formula forb(m) is true. The key trick in the proof is a one-to-one correspondence between order-preserving selfmappings ofY and pairs consisted of a partition ofY and a strictly increasing mapping of a subfence ofY toY.     

A proof of Shelah's partition theorem

A self contained proof of Shelah's theorem is presented: If is a strong limit singular cardinal of uncountable cofinality and 2 > ⁺ then .     

Minimal coupled diffusion process

An attempt is made to classify the boundary of the coupled operator and to construct the corresponding minimal semigroup and minimal coupled diffusion process. The sample properties and the conservative conditions of the process are discussed also.     

An Inverse Spectral Problem for a Nonsymmetric Differential Operator: Uniqueness and Reconstruction Formula

We consider an eigenvalue problem for a system on [0, 1]: $$\left\{ {\begin{array}{*{20}l} {\left[ {\left( {\begin{array}{*{20}c} 0 & 1 \\ 1 & 0 \\ \end{array} } \right)\frac{{\text{d}}} {{{\text{d}}x}} + \left( {\begin{array}{*{20}c} {p_{11} (x)} & {p_{12} (x)} \\ {p_{21} (x)} & {p_{22} (x)} \\ \end{array} } \right)} \right]\left( {\begin{array}{*{20}c} {\varphi ^{(1)} (x)} \\ {\varphi ^{(2)} (x)} \\ \end{array} } \right) = \lambda \left( {\begin{array}{*{20}c} {\varphi ^{(1)} (x)} \\ {\varphi ^{(1)} (x)} \\ \end{array} } \right)} \\ {\varphi ^{(2)} (0)\cosh \mu - \varphi ^{(1)} (0)\sinh \mu = \varphi ^{(2)} (1)\cosh \nu + \varphi ^{(1)} (1)\sinh \nu = 0} \\ \end{array} } \right.$$ with constants $$\mu ,\nu \in \mathbb{C}.$$ Under the assumption that p₂₁, p₂₂ are known, we prove a uniqueness theorem and provide a reconstruction formula for p₁₁ and p₁₂ from the spectral characteristics consisting of one spectrum and the associated norming constants.     

The emission spectrum of IBr excited in the presence of Argon

Wavelengths and wavenumbers of the band heads in the region 3915–3540 Å are recorded as obtained from the measurements of the plates taken on a first order 21-feet grating spectrograph. Earlier workers recently reported 40 bands of this system covering the region 3900–3800 Å. All the bands of this system obtained in the present experiments are analysed as involving the³ Π (1) state for lower state. The constants for the lower state are such that they represent well the ΔG (v+1/2) values obtained in the present experiments fromv=0 tov=26 as well as those obtained by Brown fromv=9 tov=43. The vibrational constants of the two states involved are:

$$\begin{gathered} \begin{array}{*{20}c} {\omega _e ^{\prime \prime } } \\ {137 \cdot 8 cm.^{ - 1} ,} \\ \end{array} \begin{array}{*{20}c} {\omega _e ^{\prime \prime } x_e ^{\prime \prime } } \\ {0 \cdot 571 cm.^{ - 1} } \\ \end{array} \begin{array}{*{20}c} {\omega _e ^{\prime \prime } y_e ^{\prime \prime } } \\ { - 0 \cdot 1156 cm.^{ - 1} } \\ \end{array} \begin{array}{*{20}c} {\omega _e z_e ^{\prime \prime } } \\ {3 \cdot 09 \times 10^{ - 3} cm.^{ - 1} } \\ \end{array} \hfill \\ \begin{array}{*{20}c} {\omega _e ^{\prime \prime } t_e ^{\prime \prime } } \\ { - 2 \cdot 5 \times 10^{ - 5} cm.^{ - 1} ,} \\ \end{array} \begin{array}{*{20}c} {\omega _e ^\prime } \\ {90 \cdot 1 cm.^{ - 1} ,} \\ \end{array} \begin{array}{*{20}c} {\omega _e ^\prime x_e ^\prime } \\ {0 \cdot 15 cm.^{ - 1} } \\ \end{array} \hfill \\ \end{gathered} $$     

On the Order of Magnitude of the Divisor Function

Let D be an increasing sequence of positive integers, and consider the divisor functions： d（n, D） =∑d｜n,d∈D,d≤√n1, d2（n,D）=∑[d,δ]｜n,d,δ∈D,[d,δ]≤√n1, where [d,δ]=1.c.m.（d,δ）. A probabilistic argument is introduced to evaluate the series ∑n=1^∞and（n,D） and ∑n=1^∞and2（n,D）.     

An explicit formula for |FM(1+1+n)|

We show that the number of elements in FM(1+1+n), the modular lattice freely generated by two single elements and an n-element chain, is 1 $$\frac{1}{{6\sqrt 2 }}\sum\limits_{k = 0}^{n + 1} {\left[ {2\left( {\begin{array}{*{20}c} {2k} \\ k \\ \end{array} } \right) - \left( {\begin{array}{*{20}c} {2k} \\ {k - 2} \\ \end{array} } \right)} \right]} \left( {\lambda _1^{n - k + 2} - \lambda _2^{n - k + 2} } \right) - 2$$ , where \(\lambda _{1,2} = {{\left( {4 \pm 3\sqrt 2 } \right)} \mathord{\left/ {\vphantom {{\left( {4 \pm 3\sqrt 2 } \right)} 2}} \right. \kern-0em} 2}\) .     

An Upper Bound for the Cardinality of an <Emphasis Type="Italic">s</Emphasis>-Distance Set in Euclidean Space

In this paper we show that if X is an s-distance set in ^m and X is on p concentric spheres then Moreover if X is antipodal, then .     

Rate-of-convergence bound for difference schemes for the system of equilibrium equations of a rigidly clamped nonhomogeneous anisotropic elastic solid

Exact difference scheme operators are used to construct a difference scheme for a system of equilibrium equations of a rigidly clamped nonhomogeneous anisotropic elastic solid. The rate-of-convergence bound .Translated from Vychislitel'naya i Prikladnaya Matematika, No. 62, pp. 14–19, 1987.     

How to find many counterfeit coins?

We propose an algorithm for findingm defective coins, that uses at most + 15m weighings on a balance scale, wheren is the number of all coins.     

On some new identities for the Fibonomial coefficients

Let F _n be the nth Fibonacci number. The Fibonomial coefficients \(\left[ {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right]_F\) are defined for n ≥ k > 0 as follows $$\left[ {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right]_F = \frac{{F_n F_{n - 1} \cdots F_{n - k + 1} }} {{F_1 F_2 \cdots F_k }},$$ with \(\left[ {\begin{array}{*{20}c} n \\ 0 \\ \end{array} } \right]_F = 1\) and \(\left[ {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right]_F = 0\) . In this paper, we shall provide several identities among Fibonomial coefficients. In particular, we prove that $$\sum\limits_{j = 0}^{4l + 1} {\operatorname{sgn} (2l - j)\left[ {\begin{array}{*{20}c} {4l + 1} \\ j \\ \end{array} } \right]_F F_{n - j} = \frac{{F_{2l - 1} }} {{F_{4l + 1} }}\left[ {\begin{array}{*{20}c} {4l + 1} \\ {2l} \\ \end{array} } \right]_F F_{n - 4l - 1} ,}$$ holds for all non-negative integers n and l.     

Коэффициенты рядов Ф урье и класс Липшица

We generalize and sharpen certain results concerning Fourier series from the Lipschitz class. In particular, for sinnx we prove the following: Let ¦b_n¦n^–2L(n) where L(x) is a continuous and slowly oscillating function. Then